Gorenstein-projective and semi-Gorenstein-projective modules

TL;DR

This paper investigates conditions under which semi-Gorenstein-projective modules are actually Gorenstein-projective, providing new criteria and examples that clarify the relationship between these classes of modules.

Contribution

It establishes conditions for an algebra to be left weakly Gorenstein, ensuring semi-Gorenstein-projective modules are Gorenstein-projective, and provides a counterexample illustrating independence of reflexivity conditions.

Findings

Finiteness of certain modules implies algebra is left weakly Gorenstein

Constructed a 6-dimensional algebra with a semi-Gorenstein-projective module not torsionless

Demonstrated independence of total reflexivity conditions of Avramov and Martsinkovsky

Abstract

An A-module M will be said to be semi-Gorenstein-projective provided that Ext^i(M,A) = 0 for all i > 0. All Gorenstein-projective modules are semi-Gorenstein-projective and only few and quite complicated examples of semi-Gorenstein-projective modules which are not Gorenstein-projective have been known. The aim of the paper is to provide conditions on A such that all semi-Gorenstein-projective modules are Gorenstein-projective (we call such an algebra left weakly Gorenstein). In particular, we show that in case there are only finitely many isomorphism classes of indecomposable left modules which are both semi-Gorenstein-projective and torsionless, then A is left weakly Gorenstein. On the other hand, we exhibit a 6-dimensional algebra with a semi-Gorenstein-projective module M which is not torsionless (thus not Gorenstein-projective). Actually, also the dual module M* is semi-Gorenstein-projective module. In this way, we show the independence of the total reflexivity conditions of Avramov and Martsinkovsky, thus completing a partial proof by Jorgensen and Sega. Since all the syzygy-modules of M and M* are 3-dimensional, the example can be visualized quite easily.